A math problem solving strategy that's proven to work

As I said before, there's a tough set of requirements that any problem solving strategy that's worth teaching to students as a singular go-to overarching framework must satisfy. I have yet to see a simple problem solving strategy satisfy all of those requirements, but here below is something that's been proven to work [1].

The Formulaic Action Oriented Problem Solving Strategy

It's "formulaic" because the back-end of the strategy is centered around using formulas to derive numeric solutions. But also, this strategy is "formulaic" because it's step-by-step, bringing students from start to finish of solving a problem.

It's "action oriented" because it critically answers for students the question of "what should I do?"

Many math problem solving strategies are problematic: here's why

(To see what a better problem solving strategy would look like, skip to the bottom... )


Many problem solving strategies in math are problematic in themselves.  Here are some typical problems with them:


(1) Some are too high level or abstract, and thus provide little to no traction for students to get into practically solving problems. Students either don't know what exactly they should do, or they misunderstand the intent of a step in the strategy.

e.g. one strategy calls for the student to describe any "barriers or constraints that may be preventing them from achieving their goal".  Would "the teacher hasn't taught us" be a valid barrier?  (The answer could be "yes", by the way.)


(2) It assumes a pretty smart student, rather than assume a barely passing one.  So the strategy works great for students who would've done fine without your help in the first place.

Different Types of Learning Tasks and Content Delivery Styles

The following lists a number of learning tasks students may engage in by teacher instruction, and thus they are instruments of teaching a teacher may employ.  Notes regarding each are provided briefly to highlight a range of possible tools teachers may use to engage students in learning.  These are arranged generally speaking from easiest to most difficult to employ effectively.

Every single one of these types of tasks and styles has its rightful place in a course --- there is no magic bullet solution.  Exclusive use of any single style or type of task is not necessarily a good thing as there are always trade-offs to be made.  This list is not meant to be a complete or exhaustive listing of all possible learning tasks or their categorization or description, but a way of thinking about them to help create an effective mixture of learning tasks for any course of learning.

Lectures
- teacher-oriented learning task
- active instrument
- students may remain largely passive for task completion, and can passively wait it out

Theory and verbal/textual explanations are delivered by an instructor to students (focuses on "telling" over "showing").  The delivery is often best done scripted and in an organized manner, almost like a audio/visual version of a printed book delivery of theory or textual explanations.  This term describes a type of teacher-oriented activity.

New angle on math education: lessons from product design

Or a user interface for learning math

High school is when many students typically first encounter some of the greatest difficulties in learning what forms the basis for learning post-secondary level math. Those are, therefore, some very critical years in a student's math educational career. If it is important to get more students to consider entering a STEM career (and I'd argue it is [1]), then it is important to consider how math can be taught to these students so they'd be more interested in it, and be better at it later on.

The following is a long piece discussing the problems facing designing a better math educational experience, complete with the principles of design that can be used to solve such problems, closing with some guidelines for how to create a better course.  In principle, all of the following can apply equally well to other subjects, like computing science, or English, but math education is tough, so I’m picking on the hardest as an example.

Math course as a product

Just how do students experience of learning math get formed? Rather than looking at it in a traditional teacher's mindset, let's look at it in terms of a commercial product designer's mindset. That means we see students as interacting with a complete, full-fledged math education product that promises students that they'd get what they want.

Complex numbers, exponentials, and the 2D plane: geometric and physical intuitions leading to Euler's Formula

**And why polar coordinates turns out to be really useful, and not just for tracking down polar bears.**

Today's goal is to re-imagine the two-dimensional plane, 
to develop a better understanding of how the 2D plane
relates to complex numbers and the complex exponential functions.

We'll do so using mainly our physical and geometric intuitions,
with some basic high school math as our only support.

So we'll need nothing more than high school algebra: i.e.
rules of exponents (i.e. Sum of Exponents rule),
rules of logs (i.e. Sum of Logs rule),
knowing how to use the exponential and 
logarithmic functions with real numbers,
knowing basic trigonometry around the unit circle,
and maybe a little about how angles are measured in radians.

There's no need to know anything about complex 
numbers or vector math.  In fact, we don't even need to know anything about
quadratic equations and how calculating its roots can give us imaginary numbers.

Best of all, by the end, **we're going to re-discover Euler's Formula and 
Euler's identity using only our physical and geometric intuitions** --- no power 
series, and no calculus.

Let's start!

## Grid View: Coordinates

The usual way to think of the 2D plane so that we can give every point
on the plane an "address" is to draw the x- and y-axis, then draw grid
lines everywhere.  This is the **grid system** that we'll start with.

![Standard cartesian plane grid system with x and y axis](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3g6uNYM5hpFLcYf9hG9cZ1RNN6q6UQIKGA902iGtaubcEKKTEUEczU1T5Z3kPGdsLX9FgUELGUfFGykdOQUCuUUimSQlKVLE43c6cRFUpV5sRQ1nv7EZ1yuzzsWHiltBTBuLH2ZCOpEw/s1600/1_grid.txt.png)

Then every point can be addressed by a rectangular coordinate
\\((x,y)\\) as usual.  If all we know is standard grade-school math,
then we can't really do much with these coordinates.  That means there's no
arithmetic of coordinates in standard grade-school, i.e. so we can't
add or subtract coordinates from each other.

Rectangular coordinates are perfectly fine for giving each point on the
2D plane an address though.

![Standard cartesian plane grid system gives every point an x and y coordinate component](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvSPWV2REoaUOoB5zK9EClhwaShc8Anbm8j3FUq3vZeJFn_potwDTCMMtKjPWcQTW_L_9Ri9HJWMLaOpmttaYcNGHU631KcmwHTZZryxuQ-VaSXgkndyqtmBti-NYKC2ExlKM2NMVhiZM/s1600/2_xy_pt.png)


## Rectangular View: Right and Up

We can **create** a kind of arithmetics of coordinates though
if we adopt a geometric view of the 2D plane rectangularly:

- Define the **rt** unit of measure as the ruler pointing 
  rightward of length 1.

- Define the **up** unit of measure as the ruler pointing
  upward of length 1.

![Define rt and up as units of rectangular view measures in the grid](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqGvFpuFwB8Ap6eO04TWrvhOk2tPAbGRzqbtj23F8HUoNp1CYhZpBEMwhwFQk55G1xvEcbV-iAMmsCf3CHsgzg8-oKQV45xa0TjnijV5pXe-bGMvnZKgTGrwmO6cgrBhrId1bKxxtg4fg/s1600/3_new-units-rt-up.png)

Then every point can be addressed by starting at the
origin \\((0,0)\\), and going a certain multiple of **rt**
plus going a certain multiple of **up**.  I.e.:

$$(x,y) = x \cdot\mathrm{rt} + y \cdot\mathrm{up}$$

![Example of adding 3 rt and 2 up to get coordinate at x equals 3 and y equals 2](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJpf-so6MDiBcEMbb5sTp_elqlDucZ7TEqgNRWcNALvqMYJRMb_hFzkCxleioEj1jUWQX81nnu0DPmxU-257Ix5_TSemi7_l0q0HweAVFjO7QYWmVTtGsZLxkJcfxkCqco6iLGTEJPCNQ/s1600/4_example_rt_up_addition.png)

That is an abuse of the "+" symbol to mean "go one direction,
plus go another direction", but that's what we're trying to
do: make arithmetic sense out of things that doesn't 
naturally have arithmetic meaning.

Notice though that the same point in the same spot of the 2D plane
now has two ways of being addressed!  The grid view gave us **coordinates**, vs.
the rectangular view that gave us an arithmetic **sum** of the **rt** and **up**
units.  We now know the two different addresses are "equal" only because they
are addressing one and the same point in the 2D plane!

We haven't, however, defined a multiplication symbol for mixing
the two units **rt** and **up** together yet,
because what could it possibly mean to multiple a **right**
with an **up** direction?

Computer Science: a diluted term, soon to be meaningless

I just saw a grade 6 school teacher's course web page.  He teaches a "Computer Science" course wherein students learned to type and format letters in Microsoft Word and Apple Pages.

There's a high school teacher who teaches "Computer Science".  His course emphasizes drawing and painting in Photoshop, and animating things in Flash.  Yes, I know you can do a lot of programming in Flash.  No, they don't do very much programming in that course.

Teachers: Please, don't call typing and formatting a letter in a word processor "Computer Science". Please don't call a multimedia class "Computer Science".  It's confusing students into thinking "Computer Science" is what used to be called "Computer class".

If your class has a mix of subject matters, call it "Computer class", then within it you can have units on "Business Applications" or "Information Processing", "Multimedia", and maybe even a unit on "Computer Science" (which would only have programming and/or the more "math side" of computer science).

Otherwise, students will one day step foot in a real Computer Science class, be it in a high school with a real CS focused teacher, or in college or university.  And they're going to go "WTF, this isn't CS", when in fact they were duped in earlier grade school.

I've seen this happen before in a different subject: engineering.  Schools started offering "engineering" courses that were really just the old electro-technologies courses, which were closer to engineering-technician courses.  Imagine the surprise when some students who took those courses and got into university engineering then faced down the barrage of engineering math and physics courses.

Actually, who am I kidding.  It's already happening with Computer Science.  The term is meaningless among many junior high students.  Many of them think it's Photoshop, PowerPoint, or photography.

This is brand dilution.  And it's too late.

Worries over losing deep conceptual knowledge: Better teaching in any subject, part 4

• This is part of the "Better teaching in any subject" series:
• Part 1: Problems with modern inquiry based methods
• Part 2: Meaning is use
• Part 3: Facilitate learning through the inner game of meaning as uses
• Part 4: Worries over losing deep conceptual knowledge

A concern some may have with facilitating learning through the inner game of meaning-as-uses is that it seems to turn everything into decomposed techniques and skills, lacking in holistic, deep, conceptual, or otherwise "meaningful" knowledge.

A moment's thought should give you comfort that that's very far from the truth.

Imagine complaining that by breaking tennis up into the various forehand and backhand techniques, players will lose sight of the holistic meaningful concepts required to understanding tennis.

I'd imagine a lot of the worry comes about from not "seeing" the holistic concepts being taught or learned when seeing only the individual techniques being learned.  Traditionally, we'd see the worksheets for practicing factoring quadratics but never see the worksheets for learning what quadratics are conceptually, or what the meaning of factoring is.  From that, we might be led to believe that there must be something wrong with worksheets or with not teaching concepts and meanings (as if we could even directly teach concepts or meanings at all).

Facilitate learning through the inner game of meaning as uses: Better teaching in any subject, part 3

• This is part of the "Better teaching in any subject" series:
• Part 1: Problems with modern inquiry based methods
• Part 2: Meaning is use
• Part 3: Facilitate learning through the inner game of meaning as uses
• Part 4: Worries over losing deep conceptual knowledge (forthcoming)

The inner game of meaning: a lesson from tennis

A lesson from the Inner Game of Tennis (Gallwey) we might draw from is that consciously and intellectually solving the problem of "what is the instructor doing" is like a fool's errand. Because the actual problem the student is trying to solve is "how do I [the student] hit that tennis ball in that situation".  Solving the former problem may help with solving the latter, but there is no guarantee of effectiveness or efficiency.

Because of the unique cognitive and physical characteristics of each student, the solution to the actual problem is always unique anyway.  It always require each student to solve it anew.  The instructor can only point in a general direction, but the student has to go the final distance to arriving at a personalized solution.

If a student's energy is devoted to solving the problem of "what is the instructor doing", then the student will have little energy left for what is more important: solving the actual problem anew for themselves in a way that fits their own unique cognitive and physical characteristics.

How to facilitate learning proper meanings from proper uses

"Meaning is use" means that meaning comes from a variety of particular uses, and students need to look and see while teachers show the varieties of uses properly in order to learn the proper meanings from their proper uses.  But because every student brings with them a different set of prior learning and experiences, a way of conceptual thinking (and physical doing) that works for one student may not work for another.

Meaning is use: Better teaching in any subject, part 2

• This is part of the "Better teaching in any subject" series:
• Part 1: Problems with modern inquiry based methods
• Part 2: Meaning is use
• Part 3: Facilitate learning through the inner game of meaning as uses
  (forthcoming)
• Part 4: Worries over losing deep conceptual knowledge (forthcoming)

Meaning is use

Meaning as use is a philosophical concept of meaning from Wittgenstein:

"For a large class of cases of the employment of the word 'meaning' --- though not for all --- this way can be explained in this way: the meaning of a word is its use in the language" (Philosophical Investigations).

Many traditional and folk understanding of meaning explains meaning in terms of mental representations, or idealized objects in some (sometimes mathematical) objective space, etc. --- i.e. stuff in people's heads or in some Platonic ideal space that has no practical significance for teachers in the classroom.

So while we may not necessarily agree that meaning is philosophically just its use in the language, it's certainly practical to see it that way for teaching!  Because we can, as Wittgenstein urges, look and see the variety of cases in which a word is used, but we cannot look into the heads and minds of students --- and more importantly, nor can students look into the minds of teachers in learning what the teacher meant.

"So different is this new perspective that Wittgenstein repeats: 'Don't think, but look!' (PI 66); and such looking is done vis a vis particular cases, not generalizations." [1]

Meaning as discussed usually refers to meaning of words in a language, but math is no different.  Math is itself a natural language, with a grammar and semantics that's evolved in the mathematical community, used to talk about things and their relationships.  We need not look further than many science research papers wherein authors write mathematical notations and formulas, interweaved with English prose, to see how math is very much a language we can talk about things with.

If we accept that meaning (of words) comes from their use, then the meaning of a thing like a math formula is also built up from the use of it.  The proper meaning of a math formula doesn't come from the instructor explaining it, and it doesn't even come from students discussing and talking about it.  The meaning of a math formula comes from the proper use of it.

Problems with modern inquiry based methods: Better teaching in any subject, part 1

• This is part of the "Better teaching in any subject" series:
• Part 1: Problems with modern inquiry based methods
• Part 2: Meaning is use (forthcoming)
• Part 3: Facilitate learning through the inner game of meaning as uses
  (forthcoming)
• Part 4: Worries over losing deep conceptual knowledge (forthcoming)

Problems with experiential, discovery, inquiry, and constructivist learning and teaching

In education, teachers nowadays are often taught constructivism and other modern inquiry based teaching and learning methods.  Those teaching methods purport to help educators teach children in a way that helps the kids construct their own meaning of what they are to learn.  One of the central claims is that meaning is constructed through experiencing, and reflecting on those experiences, on the basis of concepts and meanings learned previously.

By "modern inquiry based" teaching and learning methods, I mean the constellation of academic philosophies and folk understandings of experiential, discovery, inquiry, and constructivist learning and teaching methods.

What's frustrating is that the core understandings in modern inquiry based teaching and learning methods are not so much as wrong, but are just not very helpful to teachers.  Not helpful because the core pedagogical ideas basically only tell teachers that kids must learn from experiences and reflection.  Since we're not privy to see or control all the stuff that happens in the kids' heads anyway, therefore all the philosophically interesting parts of constructivism have no practical significance in the classroom.

Reasons to learn math you'll never use

The film, "An Education", is about the experience of a teenaged girl (Jenny) growing up, meeting an older man, and making decisions regarding pursuing school versus marriage.  There is a passage near the end of the film that rather haunts me, especially the very last line (in bold below):

Jenny: Studying is hard and boring. Teaching is hard and boring. So, what you're telling me is to be bored, and then bored, and finally bored again, but this time for the rest of my life? This whole stupid country is bored! There's no life in it, or color, or fun! It's probably just as well the Russians are going to drop a nuclear bomb on us any day now. So my choice is to do something hard and boring, or to marry my... Jew, and go to Paris and Rome and listen to jazz, and read, and eat good food in nice restaurants, and have fun! It's not enough to educate us anymore Ms. Walters. You've got to tell us why you're doing it.

An Education

It strikes me that much of the frustration at all levels of education comes down to people having varying beliefs that are often contradictory, sometimes confused, and at other times missing altogether, about the purpose of education (and by that, I mean formal education in the school setting).

Some think education is for opening the mind of young people to the light of reason.  Some believe education is essentially career and job skills training.  Some think it is to "raise the kids" in the parenting sense, as though teachers are parents.  Many think it's more or less pointless, but essential as a child caring service while the children's parents are out making a living.  Some don't think about the question altogether, opting to do whatever the teachers say, as education is just traditionally "the thing to do".

The list of purposes ascribed to education by various people goes on...

Why push math education onto students?

Math education is important for many reasons, but I will focus on offering an answer from a societal economic policy level that stresses employment and the labour market.  I will refrain from mentioning all the other great reasons for having a great math education for children in this post.

This all came to my mind after reading Ramanathan's article in The Washington Post.  Ramanathan is a professor emeritus of math, statistics, and computer science, at the University of Illinois.  He contends that although a lot of effort and money has been put into making math seem essential to every child's and adult's daily life, it is not, as compared to history, politics, or music. Further, the article contends that no one should feel obligated to love math, any more than anyone is obligated to love grammar, or composition.  He contends that there is no evidence that all the money spent has actually helped student math achievement.

There really are a number of different issues that Ramanathan is attacking.  They include:

1. Should every person love math any more than grammar or whatever else that is perhaps boring and pedantic to some?
2. Has the money spent on improving math education been spent effectively?
3. Is math really as essential to every child's and adult's daily life as politics or history?
4. Given the answers to the previous questions, should extra effort or money be spent into improving math education and to improve students' math achievement?

The last question is, insofar as education is publicly funded (it is, more of less, in Canada. I am aware the writer is talking about the USA), a matter of public policy.  So we have to put our public policy hat on before continuing, because it makes a difference in how the question is answered.

The difference is this: when it comes to public policy, the answer to the first three questions are irrelevant to the fourth.

Here's why.

Dear Grade 9s: What algebraic equations are for

A friend of mine was making an algebra test for his junior high math class recently and asked me what I thought equations are for.  What he meant is that the students are required to write out an equation for a given word problem before solving the equation, but students are bound to ask what the point of writing an equation is if they can solve it without it.

For assessment purposes, of course teachers need to see students write out an equation to know that they can.  But for students, the problems are probably easy enough that they don't really need to write out an equation to solve the problem, and so they see no point in doing the extra algebraic work!  Of course, they have also been trained over the years to recognize nice, round numbers as "The Answer" to all types of questions, so it's hard for them to understand how an equation can be part of "The Answer."

So here's a simple reason that I might give to a hypothetical grade 9 class: the point of writing an equation is to communicate an argument, and to tell the reader what the writer thinks is in fact true reality.

Math is a very Zen activity

I was reading some computer science papers recently that used quaternion numbers. 

What are quaternions? I don't know, but that's okay. I just accept them for what they are. Let them be and flow with them. I keep reading.

But how can you understand what you're reading if you don't know what these quaternion things are? That's a good question. I guess I do by first understanding what quaternions are.

You just said you don't know what they are. That's true.

So how can you know what they are? By sitting quietly, reading the papers, observing how these quaternions behave in their natural habitat.

How they behave? Sure. They certainly interact with the authors of this paper.  The authors speak of them, use them, toss them around so to speak. They behave a certain way, and interact with others and other things.  They interact with other numbers, for example.  I observe them.  And by watching over them, I learn what they are.

And then you can understand the rest of the paper.  Yes. Now I have to read a paper on the biology of the brain. It talks about these Koniocellular, contralateral LGN, and lots of other stuff.

You have no idea what they are, do you?  Nope. Sounds like fun.